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Congresso del Dipartimento di Fisica Highlights in Physics 2005 11–14 October 2005, Dipartimento di Fisica, Università di Milano Structure, formation and dynamical evolution of elliptical galaxies S.E. Arena * , G. Bertin * , L. Ciotti † , T.V. Liseikina ^,# , F. Pegoraro $ , M. Trenti & ,

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Congresso del Dipartimento di Fisica

Highlights in Physics 2005

11–14 October 2005, Dipartimento di Fisica, Università di Milano

Structure, formation and dynamical evolution of

elliptical galaxies

S.E. Arena*, G. Bertin*, L. Ciotti†, T.V. Liseikina^,#, F. Pegoraro$, M. Trenti&,

and T.S. van Albada+

* Dipartimento di Fisica, Università di Milano

† Dipartimento di Astronomia, Università di Bologna

^ Ruhr-Universitaet, Bochum, Germania

# Institute of Computational Technologies, Novosibirsk, Russia

$ Dipartimento di Fisica, Università di Pisa

+ Kapteyn Astronomical Institute, Groningen, Olanda






(3/4; 5.0)



(3/4; 5.0)





r, a=3.1

r, a=4.9

F, a=3.1

F, a=4.9




Elliptical galaxies may be imagined to have formed from collisionless collapse, reaching dynamical equilibrium by incomplete violent relaxation. We have tested this picture by showing that analytical models constructed under the above scenario and general statistical considerations, the so-called f(n) models, not only match the basic structure of elliptical galaxies (for a constant mass-to-light ratio), but are able to fit the density profiles (over nine orders of magnitude; with relative error within

10%) and the phase space properties (predicting the pressure anisotropy profile

focusing on the slow evolution induced by dynamical friction of a host galaxy

finding in general that (i) The role of collective effects and of inhomogeneities is important; (ii) The density distribution of the host galaxy tends to relax to a broader profile, in contrast with the expectations of adiabatic models; (iii) Satellites spiraling in on quasi-circular orbits tend to heat the stellar system preferentially in the tangential directions. Finally, we are opening the way to the construction of models characterized by a significantly non-spherical geometry [3].

on a minority component of “satellites”, in a laboratory of N-body simulations. The basic mechanisms have been modeled long ago by Chandrasekhar (1943), but are not understood under realistic conditions. After a first study [2] of the evolution of an n=3 isotropic polytrope, we now address a sequence of realistic galaxy models (the f(n) models mentioned above),

with mean error of 5%) of the results of collisionless collapse in a variety of N-body simulations [1]. To our knowledge, this is the first time that an analytically simple model constructed from physical arguments is matched successfully to the results of N-body experiments of galaxy formation. We have then addressed the issue of the dynamical evolution of such stellar systems,


  • 1.B Comparaison with the products collisionless collapse
  • Code: calculates the evolution of a system of simulation particles interacting with one another via a mean field, calculated from a spherical harmonic expansion of a smooth density distribution.
  • Numerical Simulations: initial conditions are clumps uniformly distributed in space in approximate spherical symmetry and with a small value of the virial parameter u(u=-2K/W<0.2). Most simulations have been carried out with 8 105 particles. From such initial conditions, the collisionless ''gravitational plasma'' evolves undergoing incomplete violent relaxation.
  • Fits:One example is in the figures 3-5 for (n; Y)=(5/8;5.4).
    • Density profiles: the fits are satisfactory not only in the outer parts, where the density falls under a treshold value that is nine orders of magnitudesmaller than the central density, but also in the inner regions (relative error within 10%).
    • Anisotropy profiles: are represented extremely well by the models (relative error within 5%).
    • Phase Space: The final energy density distribution N(E) is in very good agreement with the models, especially for the strongly bound particles. At the deeper level of N(E,J2) simulations and models also agree very well.
  • 1.A Construction and dynamical properties
  • f() are a family of theoretical collisionless models derived by extremizing the Boltzmann entropy at fixed values of the total mass,

of the total energy and of an additional third quantity Q, defined as:

As shown in (Stiavelli & Bertin 1987), this leads to the following distribution function:

where a, A, d and  are positive real constants.

The two-parameter family of models, constructed by solving the Poisson equation, is described by the  parameter and the concentration parameter =-a(0), the dimensionless depth of the central potential well.

  • Density profiles: Some examples in figure 1.
    • r - 4 beahviour, from not too far beyond rM, the half mass radius.
    • Increasing  profiles go from a prominent core to an high central concentration.
    • Projected density profiles are well fitted by the R1/n law with the index n ranging from 2.5 to 8.5.
  • Pressure anisotropy profiles: Some examples are in figure 2. (r)=2-(<w2>+<w2>)/<wr2>, w are spherical component of the particles velocity.
    • Core isotropic and outer parts radially anisotropic.
    • Higher values of  are associated with a sharper transitionfrom central isotropy to radial anisotropy.
    • The central isotropic region increases with .

1.B Comparaison with the products of collisionless collapse

(n; Y)


(5/8; 5.4)

(5/8; 5.4)

The end products of

high resolution simulations of

galaxy formation,

where incomplete violent relaxation

of a ''gravitational plasma"

results from a collisionless collapse process, are well and in detail

described by the f() models, in spite of their simplicity and

of their spherical symmetry.

(5/8; 5.4)


  • 2.B Galaxy evolution induced by dynamical friction
  • Problem: elliptical galaxies evolve passively (as a stellar population) but can also be subject to processes of dynamical evolution; one of these is associated with dynamical friction. These issues are of general astrophysical interest (e.g., see Nipoti et al. 2004).
  • Results: the effects of dynamical friction observed on the galaxy are:
    • decrease of the central concentration, figure 8,
    • increase of the central isotropic region, figure 9,
    • for the capture of a single satellite, change inthe galaxy shape from spherical to oblate and gain of systematic rotation.
    • The effects on more concentrated galaxies are smaller.
  • 2.A The problem of dynamical friction
  • Problem: the classical theory

(Chandrasekhar 1943): is not suited to describe real systems, characterized by inhomogeneities and complex orbits. No simple analytical theory exists able to incorporate these effects. Great help in understanding the physical processes then derives from numerical simulations.

  • Results: Figure 6 and 7.
    • The satellite falls slower than expected from classical theory.
    • The Coulomb logarithm is position-dependent, lower than expected at almost all positions and does not depend on Ms, Rs, r0 and Y (in the range explored).


It is the same code as in box 1, but with the addition of one or more particles to represent one satellite or a spherical shell of satellites.

These additional particles interact directly with the galaxy particles and among them; they are modeled as Plummer spheres with radius Rs and mass Ms.

Numerical Simulations

Initial conditions are: an f(n) galaxy, with given n and Y, and one satellite (or a shell of satellites), with given Rs and Ms, in circular orbit around the galaxy centre at initial distance r0 (or a range of distances).

Most simulations have been carried out with 2.5 105 particles for the galaxy and 1, 20, or 100 satellites. From such initial conditions the satellite (or shell of satellites) slowly sinks toward the centre of the galaxy, because of dynamical friction.

The fall of the satellites is significantly modified by the collective effects and inhomegeneities associated with the host galaxy,which, in turn, evolves by decreasing its density concentration and bychanging the pressure anisotropy in the tangential directions.


There are no systematic procedures to construct galaxy models with triaxial geometry. Only a few triaxial density-potential pairs are known, one example is that of the stratified homeoids.

To generate new models we have found it useful to consider an elementary property of the asymptotic expansion for small flattening of the homeoidal density-potential pairs.

Surprisingly, this offers a device to construct, in a systematic way, new density-potential pairs with finite deviations from spherical symmetry.

An application of this method is given in figures 10 and 11, where are illustrated the isodensity (rR2/ra, a>0) and the isopotential (F) for two toroidal models.


  • Bertin, G., Liseikina, T.V., Pegoraro, F. 2003, A&A 403, 73
  • Chandrasekhar, S. 1943, ApJ 97, 225
  • Ciotti, L., & Bertin, G. 2005, A&A 437,419
  • Nipoti, C., Treu, T., Ciotti, L., Stiavelli, M. 2004, MNRAS, 355, 1119
  • Stiavelli, M., & Bertin, G. 1987, MNRAS 229, 61
  • Trenti, M., Bertin, G., van Albada, T.S. 2005, A&A 433, 57.